### Abstract

Language | English |
---|---|

Pages | 1367–1388 |

Number of pages | 22 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 79 |

Issue number | 4 |

DOIs | |

Publication status | Published - 23 Jul 2019 |

### Fingerprint

### Keywords

- magnetic resonance elastography
- elasticity
- biomechanics
- inverse problem

### Cite this

*SIAM Journal on Applied Mathematics*,

*79*(4), 1367–1388. https://doi.org/10.1137/18M1201160

}

*SIAM Journal on Applied Mathematics*, vol. 79, no. 4, pp. 1367–1388. https://doi.org/10.1137/18M1201160

**The MRE inverse problem for the elastic shear modulus.** / Davies, Penny J.; Barnhill, Eric; Sack, Ingolf.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The MRE inverse problem for the elastic shear modulus

AU - Davies, Penny J.

AU - Barnhill, Eric

AU - Sack, Ingolf

PY - 2019/7/23

Y1 - 2019/7/23

N2 - Magnetic resonance elastography (MRE) is a powerful technique for noninvasive determination of the biomechanical properties of tissue, with important applications in disease diagnosis. A typical experimental scenario is to induce waves in the tissue by time-harmonic external mechanical oscillation and then measure the tissue's displacement at fixed spatial positions 8 times during a complete time-period, extracting the dominant frequency signal from the discrete Fourier transform in time. Accurate reconstruction of the tissue's elastic moduli from MRE data is a challenging inverse problem, and we derive and analyze two new methods which address different aspects. The first of these concerns the time signal: using only the dominant frequency component loses information for noisy data and typically gives a complex value for the (real) shear modulus, which is then hard to interpret. Our new reconstruction method is based on the Fourier time-interpolant of the displacement: it uses all the measured information and automatically gives a real value of shear modulus up to rounding error. This derivation is for homogeneous materials, and our second new method (stacked frequency wave inversion, SFWI) concerns the inhomogeneous shear modulus in the time-harmonic case. The underlying problem is ill-conditioned because the coefficient of the shear modulus in the governing equations can be zero or small, and the SFWI approach overcomes this by combining approximations at different frequencies into a single overdetermined matrix--vector equation. Careful numerical tests confirm that both these new algorithms perform well.

AB - Magnetic resonance elastography (MRE) is a powerful technique for noninvasive determination of the biomechanical properties of tissue, with important applications in disease diagnosis. A typical experimental scenario is to induce waves in the tissue by time-harmonic external mechanical oscillation and then measure the tissue's displacement at fixed spatial positions 8 times during a complete time-period, extracting the dominant frequency signal from the discrete Fourier transform in time. Accurate reconstruction of the tissue's elastic moduli from MRE data is a challenging inverse problem, and we derive and analyze two new methods which address different aspects. The first of these concerns the time signal: using only the dominant frequency component loses information for noisy data and typically gives a complex value for the (real) shear modulus, which is then hard to interpret. Our new reconstruction method is based on the Fourier time-interpolant of the displacement: it uses all the measured information and automatically gives a real value of shear modulus up to rounding error. This derivation is for homogeneous materials, and our second new method (stacked frequency wave inversion, SFWI) concerns the inhomogeneous shear modulus in the time-harmonic case. The underlying problem is ill-conditioned because the coefficient of the shear modulus in the governing equations can be zero or small, and the SFWI approach overcomes this by combining approximations at different frequencies into a single overdetermined matrix--vector equation. Careful numerical tests confirm that both these new algorithms perform well.

KW - magnetic resonance elastography

KW - elasticity

KW - biomechanics

KW - inverse problem

UR - https://epubs.siam.org/journal/smjmap

U2 - 10.1137/18M1201160

DO - 10.1137/18M1201160

M3 - Article

VL - 79

SP - 1367

EP - 1388

JO - SIAM Journal on Applied Mathematics

T2 - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 4

ER -